What is pure math?

Pure Mathematics is the branch of mathematics that deals only with mathematics and how it works - it is the HOW of mathematics. It is abstracted from the real world and provides the "tool box" of mathematics; it includes things like calculus.

Applied mathematics is the branch of mathematics which applies the techniques of Pure Mathematics to the real world - it is the WHERE of mathematics; it includes things like mechanics.

Pure Mathematics teaches you HOW to integrate, Applied mathematics teaches you WHERE to use integration.